Existential TypesExistential Types

Polymorphic Type Inference and Polymorphic Type Inference and Abstract Data TypesAbstract Data Typesby Martin Oderskyby Martin Odersky

Table Of ContentTable Of Content

Existential typeExistential type Syntax of ExMLSyntax of ExML Formal syntax of ExMLFormal syntax of ExML Type inference rulesType inference rules Type inference algorithmType inference algorithm Semantic functionSemantic function

Existential TypeExistential Type [Mit88] and [Cardelli85][Mit88] and [Cardelli85] abstypeabstype complex complex withwith

creat: real creat: real →real → complex,→real → complex, plus: complex → complex, plus: complex → complex, re: complex → real, re: complex → real, im: complex → real im: complex → realisispackpack real real ∧ ∧ realreal λλx: real. x: real. λλy: real. <x, y>y: real. <x, y> λλz: real z: real ∧∧ real. real. λλw: real w: real ∧∧ real. real. <fst(z) + fst(w), snd(z) + snd(w)> <fst(z) + fst(w), snd(z) + snd(w)> λλz: real z: real ∧∧ real. fst(z) real. fst(z) λλz: real z: real ∧∧ real. snd(z) real. snd(z)to to ∃t. [(real ∃t. [(real →→ real real →→ t) t) ∧ ∧ (t (t → t) → t) ∧∧ (t → real) (t → real) ∧ ∧ (t → real)(t → real)] ]

Syntax of ExMLSyntax of ExML

typetype [args] T = K [args] T = K11 ofof ττ 11 | … | K | … | Kkk ofof ττ kk typetype KEY = Key KEY = Key ofof ‘a * (‘a -> int) ‘a * (‘a -> int) Key(3, fun x -> 5)Key(3, fun x -> 5) Key([1, 2, 3], list_length)Key([1, 2, 3], list_length) Key(3, list_length) Key(3, list_length) (* error *)(* error *) letlet (Key(v,f)) = x (Key(v,f)) = x inin f v f v letlet (Key(v,f)) = x (Key(v,f)) = x inin v v (* error *)(* error *)

Syntax of ExML (cont.)Syntax of ExML (cont.)

typetype Key = {x : ‘a; f : ‘a -> int} Key = {x : ‘a; f : ‘a -> int} letlet z = {x = 3, f = fun x -> x + 2} z = {x = 3, f = fun x -> x + 2} inin

z.f z.x z.f z.x letlet z = {x = 3, f = fun x -> x + 2} z = {x = 3, f = fun x -> x + 2} inin

z.f z.f (* error *)(* error *) letlet z = {x = 3, f = fun x -> x + 2} z = {x = 3, f = fun x -> x + 2} inin

letlet y = z y = z inin z.f y.x z.f y.x (* error *)(* error *)

Syntax of ExML (cont.)Syntax of ExML (cont.)

typetype ‘a STACK = ‘a STACK = {Self : ‘b; {Self : ‘b; Push : ‘a -> ‘b -> ‘b; Push : ‘a -> ‘b -> ‘b; Pop : ‘b -> ‘b; Pop : ‘b -> ‘b; Top : ‘b -> ‘a; Top : ‘b -> ‘a; Null : ‘b -> bool} Null : ‘b -> bool}

letlet push x s = s with {Self = s.Push x s.Self} push x s = s with {Self = s.Push x s.Self} letlet top s = s.Top s.Self top s = s.Top s.Self

Formal Syntax of ExMLFormal Syntax of ExML

Original Type Inference RuleOriginal Type Inference Rule

Type Inference Rule ExtensionType Inference Rule Extension

Original Type Inference AlgorithmOriginal Type Inference Algorithm

Type Inference Algorithm ExtensionType Inference Algorithm Extension

Semantic Function for ExpressionSemantic Function for Expression

Semantic Function for TypesSemantic Function for Types